Gravitational Algebras in Spacetime

• Gravitational algebras are algebraic frameworks, including Lie, L∞, and von Neumann algebras, that characterize the symmetries, constraints, and observables of gravitational theories.
• They are constructed by gauging finite-dimensional symmetry algebras, leading to geometric realizations in both relativistic settings (e.g., Riemann–Cartan geometry) and nonrelativistic cases (e.g., Newton–Cartan structures).
• In quantum gravity, von Neumann factors and crossed products facilitate the quantization of diffeomorphism-invariant observables, linking operator algebras to entropy and boundary symmetries.

A gravitational algebra on spacetime is an algebraic structure—typically a Lie, $L_\infty$, or von Neumann algebra—encoding the symmetries, dynamical constraints, or gauge-invariant observables associated with gravitational theories. The precise formulation depends on the geometric regime: relativistic (Poincaré), nonrelativistic (Galilean, Bargmann), or in the context of algebraic quantum gravity, where von Neumann factors capture diffeomorphism-invariant operator algebras. These algebras structure the local and global dynamics of spacetime geometry, classify possible gauge-invariant observables, and provide the backbone for quantization and the implementation of boundary conditions.

1. Algebra Gauging and Construction of Gravitational Algebras

Gravity theories can be constructed by gauging a finite-dimensional symmetry algebra $\mathfrak{g}$ (e.g., Poincaré, Galilean, Bargmann), promoting global symmetries to local, and associating gauge fields to the algebra's generators. The one-form gauge field is $A = T_I A^I_\mu dx^\mu$, with $T_I$ generators and $A^I_\mu$ connection one-forms. The curvature two-form $F = dA + A \wedge A$ decomposes into torsion and curvature sectors upon projection onto algebra components (Lian, 2021):

• Translation generators correspond to vielbeins/coframes.
• Rotation/boost generators correspond to connection one-forms.
• Vielbein postulates (covariant constancy) eliminate redundancy, solving for the affine connection.
• Setting particular curvature components to zero implements geometric constraints, such as vanishing torsion.

Specializations yield:

• Relativistic gravity: Gauging Poincaré gives Riemann–Cartan geometry, with torsion two-form $T^a_{\mu\nu}(P) = 2 D_{[\mu}e^a_{\nu]}$ and Lorentz curvature $R^{ab}_{\mu\nu}(M)$.
• Nonrelativistic gravity: Gauging Galilean produces Newton–Cartan (NC) structures with degenerate clock/spatial metrics. Central extensions yield Bargmann gravity, incorporating a $U(1)$ gauge field for mass (Lian, 2021).

2. Galilean and Bargmann Algebras in Spacetime Geometry

Gauging the Galilean algebra, with generators $\{H, P_a, G_a, J_{ab}\}$, gives a gauge field $A_\mu$ incorporating time/space coframes and rotation/boost connections. Its curvature contains temporal and spatial torsions, $R_{\mu\nu}(H) = 2\partial_{[\mu}\tau_{\nu]}$, $R^a_{\mu\nu}(P)$, as well as curvatures for boosts and rotations.

Newton–Cartan spacetimes feature a degenerate pair ($\tau_\mu$, $h^{\mu\nu}$), a vector field $v^\mu$ with $v^\mu\tau_\mu=1$, and a completeness relation. The general NC affine connection (Hartong–Obers parameterization) is:

$$\Gamma^\rho_{\mu\nu} = -v^\rho\partial_\mu\tau_\nu + \tfrac12 h^{\rho\sigma} \big(\partial_\mu h_{\nu\sigma} + \partial_\nu h_{\mu\sigma} - \partial_\sigma h_{\mu\nu}\big) + \tfrac12 h^{\rho\sigma}\left(\tau_\mu K_{\sigma\nu} + \tau_\nu K_{\sigma\mu} + L_{\sigma\mu\nu}\right)$$

$K_{\mu\nu}$ (Newton–Coriolis form) and $L_{\sigma\mu\nu}$ (spatial contortion) remain undetermined except by boost invariance.

Central extension yields the Bargmann algebra with an additional generator $M$ and curvature $R_{\mu\nu}(M)$, corresponding to a $U(1)$ gauge field $m_\mu$ that tracks mass/particle number. The extended Bargmann frame enables manifestly invariant formulations under Galilean boosts (Lian, 2021).

3. Temporal Torsion, Foliations, and Physical Structure

The temporal torsion $R_{\mu\nu}(H)$ plays a central role:

• Generic torsional NC (TNC): $\partial_{[\mu}\tau_{\nu]}\neq 0$ allows for non-integrable time, potentially breaking causality.
• Twistless TNC (TTNC): $\tau\wedge d\tau=0$ admits foliations by time slices (up to conformal factor).
• Torsionless NC (TLNC): $d\tau=0$ yields absolute, integrable time.

Imposing $R(H)=0$ leads to absolute time; weaker conditions retain a "gravitational acceleration" two-form $a_\mu = \pounds_v \tau_\mu$ in the NC geometry. This geometric structure clearly distinguishes the nonrelativistic sector from the pseudo-Riemannian signature underpinning general relativity (Lian, 2021).

4. Gravitational Algebras in Quantum Theory: von Neumann Factors and the Crossed Product

In algebraic quantum gravity, the algebra of diffeomorphism-invariant observables is captured by von Neumann factors. Gauge-invariant algebras in semiclassical gravity are realized as crossed products by modular (boost) flows. The key constructions (Chen et al., 4 Jun 2024, Cirafici, 23 Dec 2024, Faulkner et al., 1 May 2024, Cao et al., 4 Dec 2025) include:

• For black holes or cosmological horizons, starting from a Type III$_1$ QFT algebra, crossing by the modular automorphism group associated to horizon boosts yields a Type II$_\infty$ or II$_1$ factor.
• The trace on the resulting algebra is directly related to the (generalized) entropy $S_{\rm gen}=A/4G+S_{\rm QFT}$, establishing a rigorous link between operator-algebraic entropy and semiclassical gravitational entropy.
• In spacetimes with multiple extremal surfaces (e.g., wormholes), algebras are constructed for regions associated with area sum or area difference operators, often using operator-valued weights to encode phase transitions between entanglement wedges (Cao et al., 4 Dec 2025).
• The crossed product structure allows the definition of nonequilibrium steady states, entropy production, and fluctuation theorems in gravitational systems, generalizing techniques from quantum statistical mechanics to the gravitational context (Cirafici, 23 Dec 2024).

5. Universal Corner Symmetry and Boundary Algebras

Symmetries associated with codimension-2 surfaces ("corners") are organized by the Universal Corner Symmetry (UCS) algebra:

$$\operatorname{Diff}(S) \ltimes \left(GL(2, \mathbb{R}) \ltimes \mathbb{R}^2\right)$$

where $S$ is a corner in spacetime. The algebra includes:

• Surface diffeomorphisms.
• Normal-frame transformations and Weyl rescalings $\in GL(2)$.
• Normal translations.

Coadjoint orbits of this structure encompass finite-distance and asymptotic boundary symmetries, with implications for phase space stratification, boundary charges, and quantization via the orbit method. Its realization as an Atiyah Lie algebroid organizes geometric and algebraic data naturally at the boundary, underpinning both classical integrability and quantum boundary degrees of freedom (Ciambelli et al., 2022).

6. $L_\infty$-Algebras in the Formulation of Gravity

The full gauge symmetry, field equations, and Noether identities of the Einstein–Cartan–Palatini formulation are encoded in cyclic $L_\infty$-algebras. The graded vector space packages diffeomorphisms, Lorentz ghosts, vielbein, and spin connection fields, as well as antifields and Noether structures. The primary structure maps $\ell_k$ satisfy higher Jacobi/homotopy relations and are cyclic under the pairing, capturing the complete BRST–BV algebra of gravity. In $d=3$, the associated $L_\infty$ is quasi-isomorphic to that of a Chern–Simons theory, but this fails in $d=4$, reflecting the absence of local degrees of freedom in four-dimensional BF theory (Ćirić et al., 2020).

7. Classification, Extensions, and Quantum Group Structures

Higher-dimensional and extended gravitational algebras are systematically classified by double extensions and semigroup $S$-expansion (e.g., for $\mathfrak{so}(2,2)$), which supply nondegenerate invariant bilinear forms essential for constructing consistent (especially Chern–Simons) gauge theories on non-semisimple symmetry algebras (Concha et al., 2023, Matulich et al., 2019). The quantum theoretic perspective incorporates Drinfel'd double structures, $r$-matrices, and quantum group symmetries, manifesting in noncommutative spacetimes with absolute time and central-mass/gauge field extensions. Contractions of relativistic quantum doubles recover nonrelativistic (e.g., Bargmann) gravitational algebras and their representation theory (Ballesteros et al., 2014, Ballesteros et al., 2010).

---

References:

• Gravity Theories via Algebra Gauging (Lian, 2021)
• Charge Algebra in Al(A)dS$_n$ Spacetimes (Fiorucci et al., 2020)
• Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension (Matulich et al., 2019)
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes (Ballesteros et al., 2014)
• $L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity (Ćirić et al., 2020)
• Universal Corner Symmetry and the Orbit Method for Gravity (Ciambelli et al., 2022)
• Extended Kinematical 3D Gravity Theories (Concha et al., 2023)
• Gravitational algebras and applications to nonequilibrium physics (Cirafici, 23 Dec 2024)
• Gravitational algebras and the generalized second law (Faulkner et al., 1 May 2024)
• A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes (Chen et al., 4 Jun 2024)
• Gravitational Algebras with Two Areas (Cao et al., 4 Dec 2025)
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations (Ballesteros et al., 2010)